Group #21425870 — subgroups

Subgroups — #21425870

Order 7552.

144 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14273667	C₂	1	1	yes	maximal, characteristic
2	#16837440	C₂	1	1	yes	maximal, characteristic
3	#16837226	C₂	1	4	yes	maximal, complemented
4	C₂×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
5	#14273686	—	2	1	no	—
6	#14273667	C₄	1	2	yes	complemented
7	#14273667	C₂²	1	2	yes	—
8	C₂×C₅₉⋊C₁₆	C₄	1	2	yes	complemented
9	#14273661	—	2	1	no	—
10	#14273659	C₂²	1	2	yes	—
11	C₂×M₁₆×C₅₉	—	2	1	no	—
12	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	#108×C₅₉	—	2	1	no	—
14	#108×C₅₉	C₂²	1	2	yes	—
15	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
16	M₁₆×C₅₉	—	2	2	no	—
17	M₁₆×C₅₉	—	4	1	no	—
18	C₂×C₈×C₅₉	—	4	1	no	—
19	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₉	Q₈	1	1	yes	characteristic
23	C₂×C₈×C₅₉	Q₈	1	1	yes	characteristic
24	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	C₂²⋊Dic₅₉	—	4	1	no	—
26	C₄⋊Dic₅₉	—	2	2	no	—
27	C₄×Dic₅₉	—	4	1	no	—
28	C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
29	C₂³×C₅₉	SD₁₆	1	1	yes	characteristic
30	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
33	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₅₉	—	4	2	no	—
37	C₈×C₅₉	—	4	2	no	—
38	C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
40	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₅₉	—	4	1	no	—
43	C₂²×C₅₉	#106	1	1	yes	characteristic
44	C₂²×C₅₉	#106	1	1	yes	characteristic
45	C₂²×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
46	C₄×C₅₉	#106	1	1	yes	characteristic
47	C₄×C₅₉	#106	1	1	yes	characteristic
48	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
50	Dic₅₉	—	8	2	no	—
51	#2155	—	59	1	no	maximal, Sylow
52	C₂×C₅₉	—	2	1	no	—
53	C₂×C₅₉	—	2	1	no	—
54	C₂×C₅₉	—	2	1	no	—
55	C₂×C₅₉	C₂×#106	1	1	yes	characteristic
56	C₂×M₃₂	—	59	1	no	—
57	C₂×#108	D₅₉	1	1	yes	characteristic
58	#427	—	59	1	no	—
59	#365	—	59	4	no	—
60	C₅₉	#2155	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	2	1	no	—
62	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
63	#117	—	118	1	no	—
64	M₃₂	—	59	2	no	—
65	M₃₂	—	59	2	no	—
66	C₂×C₁₆	—	59	2	no	—
67	#108	—	2	1	no	—
68	#108	C₂×D₅₉	1	2	yes	—
69	#107	—	118	1	no	—
70	#106	—	59	2	no	—
71	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
72	M₁₆	—	2	2	no	—
73	M₁₆	—	4	1	no	—
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	Dic₁₁₈	1	1	yes	characteristic
76	C₂×C₈	Dic₁₁₈	1	1	yes	characteristic
77	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
78	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
79	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
80	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
81	C₄⋊C₄	—	118	2	no	—
82	C₂²⋊C₄	—	236	1	no	—
83	C₄²	—	236	1	no	—
84	C₁₆	—	59	4	no	—
85	C₂³	#11745671	1	1	yes	characteristic
86	C₂×C₄	—	236	2	no	—
87	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
88	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
89	C₂×C₄	#11745670	1	1	yes	characteristic
90	C₂×C₄	#11745670	1	1	yes	characteristic
91	C₂×C₄	#11745671	1	1	yes	characteristic
92	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
95	C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
96	C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
97	C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#14273658	1	1	yes	characteristic
100	C₂²	#14273658	1	1	yes	characteristic
101	C₂²	C₂×#11745671	1	1	yes	characteristic
102	C₄	—	472	2	no	—
103	C₄	#14273658	1	1	yes	characteristic
104	C₄	#14273658	1	1	yes	characteristic
105	C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
106	C₄	C₂×#11745670	1	1	yes	characteristic
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	C₂×#14273658	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273667

C₂

yes

maximal, characteristic

#16837440

C₂

yes

maximal, characteristic

#16837226

C₂

yes

maximal, complemented

C₂×#108×C₅₉

C₂

yes

maximal, characteristic

#14273686

—

#14273667

C₄

yes

complemented

#14273667

C₂²

yes

—

C₂×C₅₉⋊C₁₆

C₄

yes

complemented

#14273661

—

#14273659

C₂²

yes

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#108×C₅₉

—

#108×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

Q₈

yes

characteristic

C₂×C₈×C₅₉

Q₈

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₅₉⋊C₁₆

C₂×C₄

yes

—

C₂³×C₅₉

SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

C₂×SD₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

Dic₅₉

—

#2155

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₂×#106

yes

characteristic

C₂×M₃₂

—

C₂×#108

D₅₉

yes

characteristic

#427

—

#365

—

C₅₉

#2155

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

—

118

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

C₂×D₅₉

yes

—

#107

—

118

—

#106

—

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

Dic₁₁₈

yes

characteristic

C₂×C₈

Dic₁₁₈

yes

characteristic

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

236

—

C₄²

—

236

—

C₁₆

—

C₂³

#11745671

yes

characteristic

C₂×C₄

—

236

—

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

C₂×C₄

Dic₅₉⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

C₈

—

C₈

Dic₅₉⋊C₄

yes

characteristic

C₈

Dic₅₉⋊C₄

yes

characteristic

C₈

C₂×Dic₁₁₈

yes

characteristic

C₈

C₂×C₄×D₅₉

yes

characteristic

C₂²

—

C₂²

#14273658

yes

characteristic

100

C₂²

#14273658

yes

characteristic

101

C₂²

C₂×#11745671

yes

characteristic

102

C₄

—

472

—

103

C₄

#14273658

yes

characteristic

104

C₄

#14273658

yes

characteristic

105

C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

106

C₄

C₂×#11745670

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

—

110

C₂

C₂×#14273658

yes

characteristic

	subgroup	G/H
center	C₄	#14273658
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₂×#108×C₅₉	C₂
socle	C₂×C₅₉	C₂×#106
hypercenter	C₂×#108	D₅₉
hypocenter	C₅₉	#2155
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

subgroup

G/H

center

C₄

#14273658

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₂×#108×C₅₉

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

N	H ≅ G/N	type
3. #16837226	C₂	semidirect
6. #14273667	C₄	semidirect
8. C₂×C₅₉⋊C₁₆	C₄	semidirect
60. C₅₉	#2155	semidirect

H ≅ G/N

type

3. #16837226

C₂

semidirect

6. #14273667

C₄

semidirect

8. C₂×C₅₉⋊C₁₆

C₄

semidirect

60. C₅₉

#2155

semidirect

Subgroups — #21425870

Conjugacy classes of subgroups

Complements